Objectives: Understand and use the Divergence Theorem. Use the Divergence Theorem to calculate flux. Recall the alternative (flux) form of Green's Theorem that is: F. Nds = $Mdy - Ndx = == C C OM әм ON + dA = √√ div FdA дх ду R R In an analogous way, the Divergence Theorem gives the relationship between a triple integral over a solid region Q and a surface integral over the surface of Q. In the statement of the theorem, the surface S is closed in the sense that it forms the complete boundary of the solid Q. In order to invoke the Divergence Theorem, you need two conditions to be met: 1.) 2.)

Use hyperbolic functions to parametrize the intersection of the surfaces x² - y² = 4 and z = 5xy. (Use symbolic notation and fractions where needed. Use hyperbolic cosine for parametrization x variable.) x(t) = y(t) = z(t) =

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Find centroid of shape

Find the area of the surface x² -6y-2z = 0 that lies above the triangle bounded by the lines x= 1, y=0, and y = 3x. The surface area is (Type an exact answer, using radicals as needed.)

Use Stokes's Theorem to evaluate Jc F• dr. Use a computer algebra system to verify your results. C is oriented counterclockwise as viewed from above. F(x,y,z) = xyzi + yj + zk, x² + y² < a² S: the first-octant portion of z = x2 over x2 + y2 = a²

Fred the snail is crawling in an annulus whose inner radius is a and whose outer radius is b. P and Q are points on the inner circle and the outer circle that move such that P, Q and the origin are always collinear. Fred crawls so that his x coordinate is the same as Q's and his y coordinate is the same as P's. Find the coordinates (x, y) of P and Q in terms of 0, b and a. a Eliminate 0 to find the Cartesian equation of Fred's path.

Represent the ellipsoid E: x? z? 1 16 + + 25 parametrically and write out the integral for its surface area A (E). (Do not evaluate the integral.) (Use symbolic notation and fractions where needed.) A(E) = " 2x død0

Find an expression for a unit vector normal to the surface x = 10 sin (v) , y = u, z = 10 cos (v) at the image of a point (u, v) for 0

段階的に解決し、 人工知能を使用せず、 優れた仕事を行います ご支援ありがとうございました SOLVE STEP BY STEP IN DIGITAL FORMAT DON'T USE AI DON'T USE AI DON'T USE AI DON'T USE AI C. Find the surface area generated by the rotation of the curve around the indicated axis. x=4t y= 3t+3 2≤t≤7 x-axis